TITLE: How to prove that $\frac{r}{R}+1=\cos A+\cos B+\cos C$?
QUESTION [10 upvotes]: How do we prove that for any triangle this holds: $$\frac{r}{R}+1=\cos A+\cos B+\cos C$$ I can use this beautiful identity to prove several geometric inequalities, but I have no idea how to prove the identity itself. Can anyone give me hints?

REPLY [5 votes]: This is corollary of Carnot theorem, which is stated that $d(O,BC)+d(O,CA)+d(O,AB)=R+r$. Let $M,N,P$ be the midpoints of $BC,CA,AB$. Then $\cos A+ \cos B+\cos C=\dfrac{OM+ON+OP}{R}=\dfrac{R+r}{R}=1+\dfrac{r}{R}.$